-2
$\begingroup$

How can I determine whether an ambiguous triangle has has one answer, two answers, or none? I understand that an ambiguous triangle is that which presents two sides and one angle (SSA), I just don't understand how to know whether it has one answer, or two.

$\endgroup$

closed as unclear what you're asking by Adam Hughes, Matt Samuel, Chappers, Daniel, graydad May 20 '15 at 0:52

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

2
$\begingroup$

Suppose that $A$ is the side adjacent to the angle, and $B$ is not. Let $A$ have length $a$ and $B$ have length $b$, and let the angle be $\theta$.

If $b > a$, then there is $1$ solution for all values of $\theta$

Assuming $\theta < \frac{\pi}{2}$ and $b \leq a$:

0 solutions: $$b < a\sin(\theta)$$ 1 solution: $$b = a\sin(\theta)$$ 2 solutions: $$b > a\sin(\theta)$$

If $\theta = \frac{\pi}{2}$, then there is a degenerate solution where $a=b$.

If $\theta > \frac{\pi}{2}$, then there are no solutions where $a < b$, and a degenerate solution where $a=b$.

I should add that a good intuition for this behavior comes from trying to construct such a triangle with a compass and straightedge, starting with the angle, then edge A, then edge B.

$\endgroup$
  • $\begingroup$ It might be worth considering cases when $b \gt a$, or when $b=a$, or when $\theta$ is bigger than a right angle and $a\sin(\theta) \le b \lt a$. $\endgroup$ – Henry May 19 '15 at 20:25
  • $\begingroup$ This is not quite right. For example, if $b>a$, then there is exactly one solution. $\endgroup$ – Jack Lee May 19 '15 at 20:31
  • $\begingroup$ One should probably stipulate that $\theta < \frac\pi2$ and $b < a$; otherwise I don't think there can be more than one non-degenerate solution (if there is even one) for which the desired SSA properties hold. (I assume $\theta$ is an interior angle by the definition of SSA.) $\endgroup$ – David K May 19 '15 at 20:33
  • $\begingroup$ There are only single or degenerate solutions in the other cases, though I shouldn't have neglected them (added now). $\endgroup$ – TokenToucan May 19 '15 at 20:38
  • $\begingroup$ If $b\gt a$, there is one solution only if we assume that the triangle has non-negative angles and sides. $\endgroup$ – John Joy May 19 '15 at 21:31

Not the answer you're looking for? Browse other questions tagged or ask your own question.